The seven circles theorem revisited

• Published in: Mathematical gazette 2018-07, Vol.102 (554), p.280-301
• Main Author: SILVESTER, JOHN R.
• Format: Article
• Language: English
• Subjects: Chains; Concurrency; Theorems
• ISSN: 0025-5572; 2056-6328
• DOI: 10.1017/mag.2018.59

The circles C 1 , & , C n form a chain of length n if C i touches C i + 1 , for i = 1, & , n − 1, and the chain is closed if also C n touches C 1 . A cyclic chain is a chain for which all the circles touch another circle S , the base circle of the chain. If C i touches S at P i , then P 1 , & , P n are the base points of the chain. Sometimes there may be coincidences among the base points; in particular, if P i = P j , then the line P i P j should be interpreted as the tangent S to at P i . The seven circles theorem first appeared in [1, §3.1], and some historical details of its genesis can be found in John Tyrrell's obituary [2]. The theorem concerns a closed cyclic chain of length 6, and says that, if a certain extra condition is satisfied, then the lines P 1 P 4 , P 2 P 5 , P 3 P 6 joining opposite base points are concurrent. Here and throughout, ‘concurrent’ should be read as ‘concurrent or all parallel’, that is, the point of concurrency might be at infinity.